Q. 353 - Intermediate Algebra

Question

In the following exercises, factor by grouping. 

$m^3+m^2+m+1$

Step-by-Step Solution

Answer

The factored form of the polynomial is,

$\left(m+1\right)\left(m^2+1\right)$

Step 1. Given Information

We are given a polynomial,

$m^3+m^2+m+1$

Step 2. Factorizing the polynomial

Grouping the terms, we get

$m^3+m^2+m+1=\left(m^3+m^2\right)+\left(m+1\right)$

Factoring out greatest common factor, we get 

$m^3+m^2+m+1=m^2\left(m+1\right)+1\left(m+1\right)$

Now, again factoring out greatest common factor, we get  

$m^3+m^2+m+1=\left(m+1\right)\left(m^2+1\right)$

Step 3. Checking the factorization

Multiplying the factors,   

$$\begin{gathered} \left(m+1\right)\left(m^2+1\right)=m^3+m^2+m+1 \\ m\left(m^2+1\right)+1\left(m^2+1\right)=m^3+m^2+m+1 \\ {m}^3+m+m^2+1=m^3+m^2+m+1 \\ {m}^3+m^2+m+1=m^3+m^2+m+1 \\ LHS=RHS \end{gathered}$$

Hence the factorization is correct.